1 00:00:00,000 --> 00:00:00,320 2 00:00:00,320 --> 00:00:03,069 In the last couple of videos we saw that we can describe 3 00:00:03,069 --> 00:00:07,810 a curves by a position vector-valued function. 4 00:00:07,809 --> 00:00:11,019 And in very general terms, it would be the x position as a 5 00:00:11,019 --> 00:00:13,750 function of time times the unit vector in the 6 00:00:13,750 --> 00:00:15,259 horizontal direction. 7 00:00:15,259 --> 00:00:19,009 Plus the y position as a function of time times 8 00:00:19,010 --> 00:00:22,040 the unit victor in the vertical direction. 9 00:00:22,039 --> 00:00:25,119 And this will essentially describe this-- though, if you 10 00:00:25,120 --> 00:00:27,160 can imagine a particle and let's say the parameter 11 00:00:27,160 --> 00:00:28,620 t represents time. 12 00:00:28,620 --> 00:00:32,030 It'll describe where the particle is at any given time. 13 00:00:32,030 --> 00:00:34,630 And if we wanted a particular curve we can say, well, this 14 00:00:34,630 --> 00:00:40,219 only applies for some curve-- we're dealing, it's r of t. 15 00:00:40,219 --> 00:00:44,129 And it's only applicable between t being greater 16 00:00:44,130 --> 00:00:46,710 than a and less than b. 17 00:00:46,710 --> 00:00:48,740 And you know, that would describe some curve 18 00:00:48,740 --> 00:00:50,060 in two dimensions. 19 00:00:50,060 --> 00:00:51,120 Just me just draw it here. 20 00:00:51,119 --> 00:00:54,919 This is all a review of really, the last two videos. 21 00:00:54,920 --> 00:00:58,050 So this curve, it might look something like that where this 22 00:00:58,049 --> 00:00:59,839 is where t is equal to a. 23 00:00:59,840 --> 00:01:02,610 That's where t is equal to b. 24 00:01:02,609 --> 00:01:06,260 And so r of a will be this vector right here that 25 00:01:06,260 --> 00:01:08,440 ends at that point. 26 00:01:08,439 --> 00:01:11,170 And then as t or if you can imagine the parameter being 27 00:01:11,170 --> 00:01:13,210 time, it doesn't have to be time, but that's a convenient 28 00:01:13,209 --> 00:01:15,299 one to visualize. 29 00:01:15,299 --> 00:01:18,030 Each corresponding as t gets larger and larger, we're just 30 00:01:18,030 --> 00:01:20,310 going to different-- we're specifying different 31 00:01:20,310 --> 00:01:22,230 points on the path. 32 00:01:22,230 --> 00:01:24,070 We saw that two videos ago. 33 00:01:24,069 --> 00:01:27,469 And in the last video we thought about, well, what does 34 00:01:27,469 --> 00:01:31,109 it mean to take the derivative of a vector-valued function? 35 00:01:31,109 --> 00:01:34,030 And we came up with this idea that-- and it wasn't an idea, 36 00:01:34,030 --> 00:01:35,549 we actually showed it to be true. 37 00:01:35,549 --> 00:01:37,530 We came up with a definition really. 38 00:01:37,530 --> 00:01:40,980 That the derivative-- I could call it r prime of t-- and 39 00:01:40,980 --> 00:01:42,219 it's going to be a vector. 40 00:01:42,219 --> 00:01:43,939 The derivative of a vector-valued function is once 41 00:01:43,939 --> 00:01:45,329 again going to be a derivative. 42 00:01:45,329 --> 00:01:49,060 But it was equal to-- the way we defined it-- x prime of t 43 00:01:49,060 --> 00:01:54,689 times i plus y prime of t times j. 44 00:01:54,689 --> 00:01:56,879 Or another way to write that and I'll just write all 45 00:01:56,879 --> 00:01:59,159 the different ways just so you get familiar with-- 46 00:01:59,159 --> 00:02:04,739 dr/dt is equal to dx/dt. 47 00:02:04,739 --> 00:02:06,329 This is just a standard derivative. 48 00:02:06,329 --> 00:02:08,650 x of t is a scalar function. 49 00:02:08,650 --> 00:02:12,939 So this is a standard derivative times i 50 00:02:12,939 --> 00:02:17,609 plus dy/dt times j. 51 00:02:17,610 --> 00:02:20,680 And if we wanted to think about the differential, one thing 52 00:02:20,680 --> 00:02:23,770 that we can think about-- and whenever I do the math for 53 00:02:23,770 --> 00:02:25,840 the differential it's a little bit hand wavy. 54 00:02:25,840 --> 00:02:26,950 I'm not being very rigorous. 55 00:02:26,949 --> 00:02:29,854 But if you imagine multiplying both sides of the equation by a 56 00:02:29,854 --> 00:02:37,590 very small dt or this exact dt, you would get dr is equal to-- 57 00:02:37,590 --> 00:02:38,740 I'll just leave it like this. 58 00:02:38,740 --> 00:02:41,330 dx/dt times dt. 59 00:02:41,330 --> 00:02:43,765 I could make these cancel out, but I'll just write 60 00:02:43,764 --> 00:02:45,139 it like this first. 61 00:02:45,139 --> 00:02:52,949 Times the unit vector i plus dy/dt times dt. 62 00:02:52,949 --> 00:02:55,310 Times the unit vector j. 63 00:02:55,310 --> 00:02:56,509 Or we could rewrite this. 64 00:02:56,509 --> 00:02:58,569 And I'm just rewriting it in all of the different ways 65 00:02:58,569 --> 00:03:00,039 that one can rewrite it. 66 00:03:00,039 --> 00:03:09,069 You could also write this as dr is equal to x prime of t dt 67 00:03:09,069 --> 00:03:10,340 times the unit vector i. 68 00:03:10,340 --> 00:03:15,969 So this was x prime of t dt. 69 00:03:15,969 --> 00:03:20,009 This is x prime of t right there times the unit vector i. 70 00:03:20,009 --> 00:03:22,750 Plus y prime of t. 71 00:03:22,750 --> 00:03:24,400 That's just that right there. 72 00:03:24,400 --> 00:03:26,210 Times dt. 73 00:03:26,210 --> 00:03:27,980 Times the unit vector j. 74 00:03:27,979 --> 00:03:31,000 And just to, I guess, complete the trifecta, the other way 75 00:03:31,000 --> 00:03:34,719 that we could write this is that dr is equal to-- if we 76 00:03:34,719 --> 00:03:37,979 just allowed these to cancel out, then we get is equal 77 00:03:37,979 --> 00:03:44,699 to dx times i plus dy times dy y times j. 78 00:03:44,699 --> 00:03:46,629 And that actually makes a lot of intuitive sense. 79 00:03:46,629 --> 00:03:48,859 That if I look at any dr, so let's say I look at 80 00:03:48,860 --> 00:03:53,325 the change between this vector and this vector. 81 00:03:53,324 --> 00:03:57,299 Let's say the super small change right there, that is our 82 00:03:57,300 --> 00:04:02,390 dr, and it's made up of-- it's our dx, our change in x 83 00:04:02,389 --> 00:04:03,489 is that right there. 84 00:04:03,490 --> 00:04:06,350 You can imagine it's that right there times-- but we're 85 00:04:06,349 --> 00:04:08,739 vectorizing it by multiplying it by the unit vector in 86 00:04:08,740 --> 00:04:10,120 the horizontal direction. 87 00:04:10,120 --> 00:04:16,069 Plus dy times the unit vector in the vertical direction. 88 00:04:16,069 --> 00:04:17,990 So when you multiply this distance times the unit 89 00:04:17,990 --> 00:04:19,632 vector, you're essentially getting this vector. 90 00:04:19,632 --> 00:04:22,560 91 00:04:22,560 --> 00:04:24,839 And when you multiply this guy-- and actually our change 92 00:04:24,839 --> 00:04:26,529 in y here is negative-- you're going to get 93 00:04:26,529 --> 00:04:28,750 this vector right here. 94 00:04:28,750 --> 00:04:31,389 So when you add those together you'll get your change in 95 00:04:31,389 --> 00:04:33,370 your actual position vector. 96 00:04:33,370 --> 00:04:36,019 So that was all a little bit of background. 97 00:04:36,019 --> 00:04:39,159 And this might be somewhat useful-- a future 98 00:04:39,160 --> 00:04:39,800 video from now. 99 00:04:39,800 --> 00:04:43,160 Actually, I'm going to leave it there because really I just 100 00:04:43,160 --> 00:04:44,910 wanted to introduce this notation and get you 101 00:04:44,910 --> 00:04:45,720 familiar with it. 102 00:04:45,720 --> 00:04:48,770 In the next video, what I'm going to do is give you a 103 00:04:48,769 --> 00:04:53,779 little bit more intuition for what exactly does 104 00:04:53,779 --> 00:04:54,779 this thing mean? 105 00:04:54,779 --> 00:04:57,559 And how does it change depending on different 106 00:04:57,560 --> 00:04:58,540 parameterizations. 107 00:04:58,540 --> 00:05:00,780 And I'll do it with two different parameterizations 108 00:05:00,779 --> 00:05:03,109 for the same curve. 109 00:05:03,110 --> 00:05:03,199